Eisenstein cohomology

Eisenstein cohomology
Name	Eisenstein cohomology
Field	Number theory; Representation theory; Algebraic geometry
Introduced	1970s–1990s
Contributors	Robert Langlands, James Arthur, Pierre Deligne, Armand Borel, Harish-Chandra

Eisenstein cohomology is a term in the interface of number theory and representation theory describing cohomology classes on locally symmetric spaces arising from noncuspidal automorphic forms, especially Eisenstein series. It links contributions of continuous spectrum to the Betti cohomology and intersection cohomology of arithmetic quotients, and it plays a central role in comparisons between cohomological and automorphic methods associated to groups such as GL(2), SL(2), GL(n), and reductive algebraic groups over number fields like Q. The study synthesizes techniques from the trace formula of James Arthur, the spectral decomposition of Harish-Chandra, and the work on special values by Pierre Deligne and Don Zagier.

Introduction

Eisenstein cohomology originates from foundational work of Armand Borel and Jean-Pierre Serre on cohomology of arithmetic groups and later developments by Gerd Harder, Robert Langlands, and Paul Garrett. It analyzes how residual and continuous parts of the automorphic spectrum, constructed via Eisenstein series associated to parabolic subgroups such as minimal and maximal parabolics, contribute to the cohomology of locally symmetric spaces attached to groups like Sp(2n), SO(n,n), and GSpin. The topic interrelates with the Langlands program, the Arthur–Selberg trace formula, and conjectures of Bloch–Kato and Beilinson on special values of L-functions.

Definition and Construction

One defines Eisenstein cohomology by considering an arithmetic locally symmetric space X_G = Γ\G/K for a reductive group G over a number field F with arithmetic subgroup Γ and maximal compact K, and studying the inclusion of spaces of automorphic forms: cuspidal forms, residual spectrum, and continuous spectrum produced by inducing cuspidal data on Levi subgroups M of parabolic P. Using the constant term and intertwining operators of Harish-Chandra and the analytic continuation of Eisenstein series of Langlands, one constructs classes in the cohomology H^*(X_G, V) for coefficient systems V coming from algebraic representations of groups such as GL(n) or SL(n). The construction leverages the theory of standard modules of Bernstein–Zelevinsky and the cohomological induction functors developed by Anthony Knapp and David Vogan.

Relation to Automorphic Forms and Eisenstein Series

Eisenstein cohomology arises concretely from Eisenstein series associated to cuspidal automorphic representations of Levi subgroups M, and the analytic properties of these series are governed by L-functions like those of Hecke, Dirichlet, and automorphic L-functions attached to Rankin–Selberg convolutions. Residues and special values of Eisenstein series correspond to poles of standard intertwining operators studied in the works of Friedrich Shahidi and Jacques Tits, with implications for the image of the Eisenstein map in the cohomology. The relationship is central to comparisons between the spectral decomposition used by James Arthur in his trace formula and the arithmetic descriptions appearing in conjectures of Deligne and Bloch on regulators.

Cohomological Results and Computations

Key results describe the decomposition H^*(X_G, V) = H^*_cusp ⊕ H^*_Eis ⊕ H^*_res where H^*_Eis denotes Eisenstein cohomology, with explicit descriptions in low-rank cases by Gerd Harder, James Franke, and Roman Bezrukavnikov. Computations use the long exact sequence for cohomology with compact supports, the Langlands decomposition of parabolic subgroups, and spectral sequences related to the Matsushima formula and Hodge decomposition on arithmetic varieties like locally symmetric spaces for GL(2), GU(n), and U(n,n). In many instances, Eisenstein cohomology classes are controlled by rationality results of Shimura and Kazuya Kato and congruences between Eisenstein series and cusp forms studied by Ken Ribet and Barry Mazur.

Applications in Number Theory and Representation Theory

Eisenstein cohomology has applications to special value formulas for L-functions such as those conjectured by Beilinson and Bloch–Kato, to the construction of Eisenstein classes in motivic cohomology used in the work of Alexander Beilinson and Goncharov, and to congruences exploited in Iwasawa theory and the proof of cases of the Main Conjecture by authors like Ralph Greenberg and Karl Rubin. It informs the classification of automorphic representations in Arthur packets for classical groups, connects with functorial lifts conjectured by Robert Langlands, and supplies explicit cohomological realizations of regulators appearing in the work of Colmez and Vladimir A. Voevodsky.

Examples and Explicit Calculations

Explicit computations have been carried out for groups such as GL(2), where Eisenstein cohomology relates to classical modular forms and Eisenstein series of Gauss and Hecke, and for GL(3) and Sp(4), where the residual spectrum yields concrete classes computed by Gerlach Harder and collaborators. In the case of Hilbert modular varieties associated to totally real fields studied by Ernst Kani and Andreas Debruyne, Eisenstein cohomology provides classes tied to partial zeta values of Dedekind and Hecke. Computations also appear in the cohomology of Picard modular surfaces related to imaginary quadratic fields investigated by Don Blasius and Jacquet–Langlands correspondences, yielding explicit relations between residues of Eisenstein series and regulators arising in Beilinson’s conjectures.

Category:Eisenstein cohomology